A twice-differentiable real or complex-valued function $f\colon U\to\mathbb{R}$ or $f\colon U\to\mathbb{C}$ , where $U\subseteq\mathbb{R}^n$ is some domain, is called harmonic if its Laplacian vanishes on $U$ , i.e. if $$\Delta f\equiv 0.$$

Any harmonic function $f\colon\mathbb{R}^n\to\mathbb{R}$ or $f\colon\mathbb{R}^n\to\mathbb{C}$ satisfies Liouville's theorem. Indeed, a holomorphic function is harmonic, and a real harmonic function $f\colon U\to\mathbb{R}$ , where $U\subseteq\mathbb{R}^2$ , is locally the real part of a holomorphic function. In fact, it is enough that a harmonic function $f$ be bounded below (or above) to conclude that it is constant.